This was 15 years, 8 months, 28 days ago

Godel's Incompleteness Theorem seems to reappear roughly a generation later in Derrida's essay "Structure, Sign and Play in the Discourse of the Human Sciences", partially in response to Claude Levi-Strauss:

...Now, ethnology -- like any science -- comes about within the element of discourse. And it is primarily a European science employing traditional concepts, however much it may struggle against them. Consequently, whether he wants to or not - and this does not depend on a decision on his part - the ethnologist accepts into his discourse the premises of ethnocentrism at the very moment when he is employed in denouncing them.

And a bit later:

It is a question of explicitly and systematically posing the problem of the status of a discourse which borrows from a heritage the resources necessary for the deconstruction of that heritage itself. A problem of economy and strategy.

Gently disregarding the terminology of economy and strategy, what we really have here is a structure working to undermine itself through the nature of its own structure. This really brings to mind Douglas Hofstadter's anecdotal explanation of Godel's Incompleteness Theorem, by giving the metaphor of a genie as a sufficiently strong axiomatic system. This genie is able to grant any wish, and so the two protagonists (the Tortoise and Achilles) ask for the wish: "I wish my wish would not be granted." This is a paradox and 'typeless wish', as Hofstadter calls it in his anecdote, and an example of an true yet unprovable construction within this axiomatic system. (And as as a result, the Tortoise and Achilles are ejected outside of the genie's world.)

Derrida talks about a 'rupture', presumably about the "destruction of the history of metaphysics". I haven't finished the essay yet, but initially he talks about a parallel between a criticism of ethnocentrism and the destruction of metaphysics, as an event that occurs when about this said 'rupture' occurs. Apparently the rupture coincides with the recognition that the center of a structure (read: the founding logic of an axiomatic system) began to be thought not as a "fixed locus but a function". This is the limit: after this, my parallels begin to break apart.

But anyways: what's interesting here is the focus onto a structure that, despite its origins, attempts to move beyond the fixed locus, and seeks to define a new center that is based on a method of function. The (dumbed-down) Incompleteness Theorem is the statement that any strong axiomatic system cannot be both consistent and complete, specifically because of the qualities of its strength. Compare this with "a discourse which borrows from a heritage the resources necessary for the deconstruction of that heritage itself".

Looking at this from a probably non-mathematical point of view, it's interesting to think of a mathematical structure that attempts to move beyond an axiomatic system, and a set of given logical laws, but instead focuses on the function and method of operation. Perhaps a system of mathematics in which the fluidity of logic is the meta-axiom itself?

And speaking of fluidity-change as a meta-axiom, a meta-rule on which the rules of the system is derived, Nomic is a great example, introduced widely by Hofstader's column in Scientific American, itself by a philosophy professor named Peter Suber. Here's the quote on wikipedia:

Nomic is a game in which changing the rules is a move. In that respect it differs from almost every other game. The primary activity of Nomic is proposing changes in the rules, debating the wisdom of changing them in that way, voting on the changes, deciding what can and cannot be done afterwards, and doing it. Even this core of the game, of course, can be changed.

- Peter Suber, the creator of Nomic, The Paradox of Self-Amendment, Appendix 3, p. 362.

later addendum: To be specific, the above (a re-centering of a structure) is what derrida is talking against, or rather, talking about in contrast to the rupture he speaks about. More on this later.